Steady state probability distribution of a two-product-type and multi-parallel-machine system under prioritized hedging point control policy

For a manufacturing system, we often need to evaluate its performance under a production control policy. The purpose of performance evaluation is to optimize the parameters in the control policy. In this paper, we investigate the method of evaluating the performance of a two-product-type and multi-parallel-machine manufacturing system. In order to evaluate its performance, the key is to obtain its steady-state probability distribution under a control policy. To solve this problem, firstly, we analyze the shape of distribution domains of state variables, which influence the steady-state probability balance equation. For one of these possible shapes, we develop the steady-state probability balance equations under the prioritized hedging point (PHP) control policy. From these equations, we obtain the marginal probability distribution for each product type for a two-product-type and multi-parallel-machine system, which has the similar form as the steady-state probability distribution of a one-product-type and multi-parallel-machine system. Then, a unified form of probability balance equation is developed from all possible shapes of distribution domains of state variables. Finally, numerical experiments are conducted to verify the correctness of the proposed method of computing the probability distribution of a two-product-type and multi-machine system.